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Unique Factorization Domains

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A Unique Factorization Domain consists of an integral domain such that for any nonzero \(x\in R\setminus R^{\ast}\) there is some unit \(u\in R^{\ast}\) and coprime irreducibles \(p_{1},\dots,p_{n}\in R\) and positive integers \(e_{1},\dots, e_{n}\) such that

\begin{equation} x=u p_{1}^{e_{1}}\dots p_{n}^{e_{n}} \end{equation}

Furthermore, this factorization is unique up to multiplication of \(u\) and \(p_{j}\)'s by units and up to permutation of indices.

Let \(R\) be a unique factorization domain, then every irreducible \(p\in R\) is prime.

If \(R\) is a UFD, then any \(x,y\in R\) has a gcd in \(R\).

If \(R\) is a UFD, then the polynomial ring \(R[x_{1},\dots,x_{n}]\) is a UFD.

1. References

  • Manuel Bronstein, Symbolic Integration 1: Transcendental Functions. Springer, second ed., 1996. Chapter 1 particularly reviews polynomial rings.
  • Nathan Jacobson, Basic Algebra I. Chapter 2.
  • Serge Lang, Algebra. Springer, third ed., GTM., 2002. Chapter 2.
  • James McIvor, Lecture Notes on Ring Theory. UC Berkeley, Math 113, Summer 2014.

Last Updated 2021-06-01 Tue 10:00.